CAS Ratemaking SeminarMarch 2005

INT-6Introduction to Profit Provision

Calculations

Ira Robbin, PhDPartner Re

Ground Rules The purpose of this session is to educate

actuaries in various methods used to compute the underwriting profit provision.

There will be no discussion of the adequacy of the premium charge for any particular consumer or particular class of consumers.

All attendees should scrupulously follow anti-trust guidelines.

Disclaimers No statements of Partner Re’s corporate

position will be made or should be inferred.

While some methods may be similar to methods promulgated by regulatory authorities, practitioners should follow actual regulatory instructions.

While some methods to be discussed are similar to methods in the Part 9 Study Note, students should consult the Study Note for exact details.

Cautions Examples are for illustrative purposes

only. Do not use the results from any example in real-world applications.

The profit load indicated from a model often depends critically on the assumptions and parameters. For ease of presentation, assumptions have been greatly simplified and hypothetical parameters have been selected.

Overview UW Profit Basics Overview of Different Methods Corporate and Regulatory Contexts Offset Formulas DCF and Risk-Adjusted DCF Single Policy Company Models Conclusion: Can They All Be Right?

Different Types of UW Profit Actual Achieved

Booked to Date vs Ultimate PY, AY, CY Direct, Gross, Ceded, Net Stat vs GAAP

Provision in Manual Rate Indicated, Filed, Approved

Provision in Charged Premium

UW Profit: Basic Equations U = P-L-X = UPM*P

X = Expense including premium tax

CR = (L+X)/P= 1- UPMUPM of –100% yields CR =200%

X = FX +VXR*PFX = Fixed expenseVXR = Variable expense ratio

P= (L+FX)/(1-VXR-UPM)

UW Profit Provision Chart

Profit Provision

Fixed Expense

Variable Expense

Loss Provision

Pre

miu

m

Examples L=50 FX=30 VXR=15% UPM = 5% P= (50 + 30)/(1-.15-.05) = 100

L=50 FX=30 VXR=15% UPM = -1% P= (50 + 30)/(.86) = 93 Note UPM can be negative!

UPM Calculation Approaches Investment Income Adjustment

CY Inv Offset and PV Differential Adequate Total Return

Ratemaking CY ROE Economic Return via Single Policy

Model IRR on Equity Flow and PVI/PVE

Economic Components DCF and Risk-adjusted DCF

Different UPM Calculation Methods 1. CY Inv Offset 2. PV Differential 3. Ratemaking CY ROE 4. DCF 5. Risk-Adjusted DCF 6. IRR on Equity Flow 7. PVI/PVE

Regulatory Context Philosophy of Regulation

State controlled vs free market approaches

Affordability and availability Solvency focus of statutory accounting

Rate Regulation Environments Prior approval/File and use/Use and file Differences by LOB and size of risk ROE vs Return on Sales

Corporate Context Pricing Analysis

UPM needed by LOB Individual large account pricing

Economic Return Net of Risk Consistency across business units Reconciliation vs actual return

Recap of UW Profit Regulation 1920’s – 1970’s: Low interest era

No consideration of investment income 5.0% UPM for most lines 2.5% for WC

1970’s – 90’s: High rate era Investment income offsets CAPM, DCF and Risk-Adjusted DCF IRR on Equity Flows and PVI/PVE

Method 1: CY Investment Income Offset (State X)

UPM = UPM0 – IIOffset UPM0 = Traditional UPM IIOffset = Investment Income Offset

IIOffset = iAT · PHSF Based on After-tax realized CY returns Actual portfolio mix of invested assets Base of PH-Supplied Funds

Policyholder Supplied Funds

Unearned Premium Balances UEPR(1-PPACQR) - RECV Net of Pre-paid Acquisition ExpenseNet of Receivables

Loss+ LAE Reserves PLR·(LRES/INCL)CY Reserves-to-Incurred Ratio PLR =Permissible Loss RatioRatio of Loss Reserve to Incurred Loss

CY II Offset- Example

UEPR 400 Earned Prem 1,000

LRES 1,200 Inc’d Loss 800

RECV 260 PPACQR 10.0%

UPM0 5.0% PLR 60.0%

After-tax Yield 4.0%

PHSF = (.4·(1-.1)-.26) + .6·1.5 =1.00

UPM = .05 - .04·1.00 = 1.0%

Method 2: Offset for PV Differential

UPM = UPM0 - PVDELLR UPM0 = Traditional UPMPVDELLR = Present Value Differential

Present Value Differential PVDELLR = PLR·(PV(X0)- PV(X))

X0 = Loss Pattern for Reference LOBX = Loss Pattern for Review LOB Interest Rate: New money after-tax

PV Differential Offset- Example PV(REF Loss Pattern) 98.0%

PV(REV Loss Pattern) 93.0%

Risk-free New Money Rate after tax

2.8%

PLR 60.0%

Traditional UPM 5.0%

PVDELLR = (.98-.93)*.60 = 3.0%

UPM = .05-.03 = 2.0%

Method 3: Ratemaking CY ROE Start with ROE equation:

EQ

TINVU

EQ

INCROE

Assume S= EQ Simplify taxes Split INV into INV on PHSF vs INV on S

Ratemaking CY ROE

ATAT i)PHSFiUPM)t1((ROE

Premium to Surplus Ratio

ROE in Ratemaking? GAAP vs Statutory

Going-concern vs Solvency Stat defined by state regulation

Calendar Yr vs Policy Yr ROE is CY Past decisions impact this CY Ratemaking is PY and prospective

Surplus in ROE Equation S = Target Statutory Surplus

S = P/ Premium-to-Surplus leverage ratio varies by LOB

Equity vs Surplus

Solve for UPM Find UPM to hit CY ROE target

)t1(

PHSFiiROEUPM ATATetargt

Ratemaking CY ROE - Example

UPM Calculation Validation

After-taxProfit

as % of P

PHSF 110.00% PHSF 4.40% 2.00 II on S 2.00%

After-tax yield 4.00% UPM -0.40%tax rate 35.00% Total 6.00%

target ROE 12.00%Surplus 50.00%

UPM -0.62% ROE 12.00%

Method 4: Discounted Cash Flow Prospective cash flow approach

founded in modern economic theory UPM = -krf +(E[rm] – rf)

k = funds generating coefficient rf = risk-free new money rate rm= market return = systematic covariance

What is Beta? Beta from CAPM

Capital Asset Pricing Model rA = rf +A(E[rm] – rf) Key CAPM Pricing Concept

Reward for taking systematic risk No reward for diversifiable risk

Beta: Covariance of Stock with Market

Applying CAPM to Insurance Insurance Betas by LOB?

Few single LOB insurance companies These don’t represent much of the market

Beta based on covariance of LOB UPM

with stock market return? Not right theoretically, but has been used CY UPMs vs PY Ratemaking – prior year CATs uncorrelated with Stock Market

DCF - Example

Risk-free rate 5.0%

Funds Generating Coefficient 1.50

Beta for LOB 1.20

E[Market yield] 10.0%

UPM = -1.50*.05+ 1.20(.10-.05) =-1.5%

Method 5:Risk-Adjusted DCF Solve for UPM so that:

)r PV(FIT, )r PV(X, )r PV(L,)r PV(P, ffAf rf = risk-free new money raterA = risk-adjusted rateFIT= income tax

Loss discounted at risk-adjusted rate

Risk-Adjusted Rate rA = rf + rm] – rf ) = Cov of liabilities with market While >0 for assets, the here is

for liabilities. Thus: <0 and rA < rf

How to get by LOB?

Simplified Taxes in RA DCF

)S on II(tPV

)r t)PV(X,-(1 )r PV(L,)t1(

)r t)PV(P,-(1

fA

f

)r1)(t1(

/Prt)r PV(X, )r PV(L,

)r PV(P,

f

ffA

f

Risk-Adjusted DCF Example

FV PV Factor DiscountedLoss (PV at RA rate) 50.00 0.95 47.50Fixed Expense 30.00 1.00 30.00Variable Expense 13.95 1.00 13.95

Subtotal 93.95 91.45

Capital 50.00Inv Inc on Capital 3.00 0.96 2.88Tax on II on Capital 1.05 0.96 1.01Pre-tax Provision 1.55

Premium 93.00 1.00 93.00

UPM -1.0% CR 101.0%

Method 6: IRR on Equity Flow Equity flow: flow of $ between an equity

investor and the insurance company Model prospective equity flows for

hypothetical insurance company writing one policy

Use accounting rules, surplus requirements, and other assumptions to derive income and surplus each time period.

EQF = INC - S

Equity Flow DiagramInvestment Income

UW Cash Flows Income Tax

Assets Liabilities and Surplus UW GainInvestables UEPR Investment Income EarnedReceivables Loss Reserves Realized Capital GainsRecoverables Expense Reserves TaxesOther Surplus Net

Equity Flows

Pool of Equity

Balance Sheet Income Statement

Single Policy Company

- +

Income and Cash Flow UW Gain = EP –IncLoss –IncExpense

Defined by accounting rules Does not depend on UW cash flows

Inv Inc = II on Invested Assets Invested Assets

Assets- Recvbl’s -Recovs Assets = Reserves + Surplus

Balance sheet must balance UW Cash flows impact Invested Assets

Single Policy Company: UW Income and Cash Flow

timeEarned Prem

Paid Premium

Inc'd Loss

Paid Loss

Inc'd Expense

Paid Expense

UW Income

0 0 50 0 0 30 16 -301 100 50 62 20 5 10 332 0 0 0 30 0 5 03 0 0 0 12 0 4 0

total 100 100 62 62 35 35 3

time UEPRLoss

ReservesExpense

Reserves Surplus

Total Liab and

Surplus Recv'bleInv'stble Assets

Inv Income

0 100 0 14 40 154 50 1041 0 42 9 10 61 0 61 5.22 0 12 4 4 20 0 20 3.13 0 0 0 0 0 0 0 1.0

Single Policy Company: Assets and Investment Income

Single Policy Company: Equity Flow and IRR

Pre-tax IRR 14.2%

timeUW

IncomeInv

IncomeTotal

IncomeChange in

SurplusEquity

Flow

0 -30 0.0 -30.0 40 -70.01 33 5.2 38.2 -30 68.22 0 3.1 3.1 -6 9.13 0 1.0 1.0 -4 5.0

total 3 9.3 12.3 0 12.3

IRR

IRR is comparable to the rate of interest on a loan

0t

tt xv0

Given flows xt , IRR is the interest rate, y, (if it exists) which solves:

1)y1(v

IRR on Equity Flows Typical EQ Flows in P/C insurance

First flow is negative Later flows are positive One sign change

IRR on EQ Flow well-defined Solve for premium to hit IRR target

Method 7: PVI/PVE Generalize ROE:

PV of INC at t=0 or t=1? PV of Balance Sheet account?

)r PV(EQB,

)r PV(INC,PVI/PVE

f

f

Equity Balance

Single Policy Company: PVI/PVE

Pre-tax PVI/PVE = 18.1%

time Income yearEquity

balance

0 -30.001 37.20 1 40.002 3.10 2 10.003 1.05 3 4.00

total 11.35 total 54.00

PV0 53.15

PV1 9.60

Chart of Methods

CY Invesment OffsetPV Loss Differential Offset

CY ROEIRR on Equity FlowsPVI/PVE

Find UPM DCFFind P Risk-adjusted DCF

Investment Income Offset

Target Return Methods Single Policy Company Model

Economic Component Methods

Conclusion No single “right” answer Use appropriate method for

situation Select parameters consistent with

method used Questions